The election in The Man Who Shot Liberty Valance represented the change from men like Tom (John Wayne) who dealt with problems with his hands or his guns and the future where men such as Jimmy Stewart (the protagonist) an attorney represented the future "legal" changes that were going occur in the West.
Why the wealthy ranchers and businessmen were so eager to "steal" the election is that was how they were going to be able to hold on to their gains. In America's Wild West, many people became wealthy through using corrupt methods such stealing land claims and cheating the rightful owners out of their stakes. To change from doing these things to stealing elections weren't seen as being major steps by those same men.